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Senior pre-university education (SPE) students experience difficulties applying mathematics to physics. This paper reports the outcome of an online explorative quantitative study of teachers' belief systems about improving transfer of algebraic skills from mathematics into physics, conducted among 503 mathematics and physics teachers working in SPE. We used a questionnaire with 16 beliefs about improving transfer, and asked teachers to select a top 5 and distribute 50 points among them. We used agglomerative hierarchical clustering to cluster qualified SPE teachers with more than 10 years of teaching experience. We found 3 large clusters, each containing naïve and desirable beliefs about transfer. These clusters turned out to be rather coherent sets of beliefs. Hence, these clusters can be interpreted as belief systems, to a certain extent justifying Ernest's [(1991). The philosophy of mathematics education. London: Falmer.] idea to cluster teachers based on their belief systems. We found relations between our groups and those of Ernest. Since naïve beliefs turn out to be weak in each cluster, science teacher educators can help science teachers to change their harmful naïve beliefs, into desirable transfer enhancing beliefs. Furthermore, we discuss some implications of our results for science teacher educators, curricula, teachers and textbooks.

Mathematics is of major importance in science subjects. Unfortunately, students struggle with applying mathematics in science subjects, especially physics. In this qualitative study we demonstrate that transfer of algebraic skills from mathematics in physics class can be improved by using pre-knowledge effectively. We designed shiftproblems involving instructional models to carry out small interventions in textbook problems. Shift-problems are feasible for teachers to adopt in teaching practice. To gain insight in the extent to which students improved their application of algebraic skills including basic skills and symbol sense behaviour, we selected three grade-10 physics students. In round one, the students solved algebraic physics problems as they appear in physics textbooks. Two weeks later in round two, the same problems were presented as shift problems to them where we activated prior mathematical knowledge by providing systematic rule-based algebraic hints at the start of these tasks. Algebraic skills were presented in a similar way to how these were learned in mathematics textbooks. We observed that students' problem-solving abilities increased from 48.5 % in the first to 81.8 % in the second round, indicating the effectiveness of how we implemented shift-problems. Furthermore, we discussed the implications of our results for the international science audience.

Students in upper secondary education encounter difficulties in applying mathematics in physics. To improve our understanding of these difficulties, we examined symbol sense behavior of six grade 10 physics students solving algebraic physic problems. Our data confirmed that students did indeed struggle to apply algebra to physics, mainly because they lacked both sufficient symbol sense behavior and basic algebraic skills. They used ad hoc strategies instead of correct, systematic rule-based procedures involving insight. These ad hoc strategies included the cross-multiplication, the numbering, and the permutation strategy. They worked only for basic formulas containing few variables. In problems with more variables, students got stuck. The latter two strategies substitute numbers for variables. The permutation strategy randomly checks several permutations to guess which one is correct. The numbering strategy substitutes numbers to check algebraic manipulations. Our results indicate insufficient focus on conceptual understanding of algebra in some mathematics textbooks, leading to reliance on poorly understood ad hoc strategies. Effective teaching of algebraic skills should not focus on either basic algebraic skills or on symbol sense behavior. Instead, both aspects should be taught in an integrated manner. Our operationalization of symbol sense behavior turned out to be very useful for analysis. In contrast to earlier qualitative studies, it provided us the opportunity to measure symbol sense behavior quantitatively. This operationalization should also be applicable to other science subjects. Furthermore, we discussed some implications of our results for curricula, teachers, science teacher educators, and textbook publishers aiming at successful application of mathematics in physics.

Students in senior pre-university education face difficulties in the application of mathematics in physics. This paper presents the results of a qualitative study on teachers' core beliefs about improving the transfer of algebraic skills to physics. Teachers were interviewed about their beliefs regarding a transfer problem from mathematics to physics for which solution algebraic skills were needed. We obtained large amount of data which were reduced to sixteen core beliefs including constraints and affordances influencing students' demonstration of coherent mathematics education (CME) and transfer of algebraic skills from mathematics into physics. These core beliefs were grouped into the five main categories 'Collaboration', 'Curricula', 'Students', 'Teachers' and 'Textbooks'. We think that our approach to pattern coding is both elegant and generally applicable to reduce code trees including large amount of data. Four core beliefs were identified as naïve beliefs, which may impede transfer. We provided a powerful remedy against such unproductive beliefs: through professional development programs teachers with such beliefs should be made aware, reflect and reconcile their naïve beliefs with those required for transfer. These core beliefs contain data to extract teachers' belief systems. Quantitative research could investigate to which extent this is the case and which beliefs these contain.